Ark. Mat., 37 (1999), 245-254
@ 1999 by Institut Mittag-Leffler. All rights reserved
Irreducibility of the punctual
quotient scheme of a surface
Geir Ellingsrud and Manfred Lehn
A b s t r a c t . It is shown that the punctual quotient scheme Qr1 parametrizing all zero-dimenOr
sional quotients OA2-~T
of length I and supported at some fixed point 0CA 2 in the plane is
irreducible.
Let X be a smooth projective surface, E a locally free sheaf of rank r>_l
on X , and let l>_1 be an integer. Let Q u o t ( E , l ) denote Grothendieck's quotient
scheme [S] t h a t parametrizes all quotients E--+T, where T is a zero-dimensional
sheaf of length l. Sending a quotient E-+T to the point ~ x c x l(Tx)x in the symmetric product St(X) defines a morphism 7r: Quot(E, 1)--+Sz(X) [8]. It is the purpose of this note to prove the following theorem.
T h e o r e m 1. The scheme Q u o t ( E , l) is an irreducible scheme of dimension
l ( r + l ) . The fibre of the morphism 7r: Q u o t ( E , l ) ~ S t ( X ) over a point ~ l~x is
irreducible of dimension ~ (rl~ - 1).
Using the irreducibility result, one can check that a generic point in the fibre
over l~cSt(X) represents a quotient E-+T, where T ~ O x , x / ( s , t t) and s and t are
appropriately chosen local parameters in OX,x, i.e. T is the structure sheaf of a
eurvilinear subscheme in X.
If r = l , i.e. if E is a line bundle, then Quot(E, l) is isomorphic to the Hilbert
scheme Hilbl(X). In this case, the first assertion of the theorem is due to Fogarty [6],
whereas the second assertion was proved by Briancon [2]. For general r_> 2, the first
assertion of the theorem is a result due to J. Li and D. Gieseker [9], [7]. We
give a different proof with a more geometric flavour, generalizing a technique from
Ellingsrud and Str0mme [5]. The second assertion is a new result for r_>2. After
finishing this paper we learned about a different approach by Baranovsky [1].
The natural generalizations of the theorem to higher dimensional or singular
varieties are false, as is already apparent in the r = l case of the Hilbert schemes [3].
Geir Ellingsrud and Manfred Lehn
246
1. E l e m e n t a r y m o d i f i c a t i o n s
Let X be a smooth projective surface and x E X . If N is a coherent (gxsheaf, then e(N~)=homx(N, k(x)) denotes the dimension of the fibre N(x), which
by Nakayama's lemma is the same as the minimal number of generators of the
stalk Nx. If T is a coherent sheaf with zero-dimensional support, we denote by
i(:/x) = h o m x (k(x), T) the dimension of the socle of T~, i.e. the submodule Soc(Tx) c
T~ of all elements that are annihilated by the maximal ideal in Ox,~.
L e m m a 2. Let [q:E---~T] E Q u o t ( E , l) be a closed point and let N be the kernel
of q. Then the soele dimension of T and the number of generators of N at x are
related as
e(N~)=i(T~)+r.
Proof. Write e(Nx)=r+i for some integer i_>0. Then there is a minimal free
resolution 0
i
~ (.~r+i
~OX,x--~vx,x
)Nx )0, where all coefficients of the homomorphism c~ are contained in the maximal ideal of OX,x. We have Hom(k(x),T~) TM
E x t , : (k(x), Nx) and applying the functor Hom(k(x),. ) one finds an exact sequence
0
, E x t , : (k(x), Nx)
, Ext~c (k(x), Ok,x) ~' , E x t ~ (k(x), Ox,x).r+i
But as a has coefficients in the maximal ideal, the homomorphism a r is zero. Thus
i. []
Hom(k(x), T)~Ext2x(k(x), Ox,x)~-k(x)
i
The main technique for proving the theorem will be induction on the length
of T. Let N be the kernel of a surjection E---~T, let x c X be a closed point, and let
)~: N---~k(x) be any surjection. Define a quotient E--~T ~by means of the push-out diagram
0
k(x)
0
"
>N
T
T
N/- -
0
~
T
T
T'
~E
NI
0.
>T
~0
>T-
70
Irreducibility of the p u n c t u a l quotient scheme of a surface
247
In this way every element (A)cP(N(x)) determines a quotient E ~ T ' together with
an element (p}eP(Soc(Ts
(Here WV:=Homk(W,k) denotes the vector space
dual to W.) Conversely, if E--+T ~ is given, any such (p) determines E---~T and a
point (A}. We will refer to this situation by saying that T ~ is obtained from T by
an elementary modification.
We need to compare the invariants for T and Tq Obviously, l(T~)=t(T)+l.
Applying the functor Hom(k(x),. ) to the upper row in the diagram we get an exact
sequence
0
, k(x)
) Soc(T')
, Soc(T )
Ext)(k(x), k(x))
k(x)
and therefore li(Tx)-i(Ts
Moreover, we have O<e(Ts
deserve closer inspection. Firstly, if e increases, then T t splits.
Two cases
L e m m a 3. Consider the natural homomorphisms f: Soc(T~)~T~-*T'(x) and
g: N ( x ) ~ E ( x ) . The following assertions are equivalent:
(1) e(T;)=e(r~)+l,
(2) (#} ~ P ( k e r ( f ) v ) ,
(3) (A} eP(im(g)).
Moreover, if these conditions are satisfied, then T ' ~ T |
and i(Ts
l.
Proof. Clearly, e (Tx~) = e (Tz) + 1 if and only if #(1) represents a non-trivial element in T ~(x) if and only if # has a left inverse if and only if A factors through E. []
Secondly, if i increases for all modifications ,~ from T to any T/, then the same
phenomenon occurs for all 'backwards' modifications #~ from T to any T - .
L e m m a 4. Still keeping the notation above, let E--+T~ be the modification of
E---~T determined by the point (A} e P ( N ( x ) ) . Similarly, for (#')EP(Soc(Tx) v) let
T,,=T/#'(k(x)). If i(7~,~)=i(Tx)+ l for all (A}eP(N(x)), then i(Tx)=i(T2,,~)- i
for all (#/)eP(Soc(T~)V) as well.
Pro@ Let
(I): Homx (N, k(x) ) --~ Homk (Ext~ (k(x), N), E x t ~ (k(x), k(x) ) )
be the homomorphism which is adjoint to the natural pairing
Homx (N, k(x))|
N)
, Extlx (k(x), k(x)).
By identifying Soc(Tx) ~ E x t ~ (k(z), N), we s e e that i(T~,x) = l+i(Tx) - rank(~5(A)).
The action of ~5(A) on a socle element #': k(x)--,T can be described by the following
248
Geir Ellingsrud and Manfred Lehn
diagram of pull-backs and push-forwards
0
>N--
>E
>T
~0
. k(x)-
- 0.
T
0
, k(x)
The assumption that i(T~,~)=l+i(T~) for all A, is equivalent to ~5=0. This implies
that for every #' and every A the extension in the third row splits, which in turn
means that every A factors through N~,, i.e. that N(x) embeds into N y (x). Hence,
for T , =E/N~, =coker(#) we get i(T,,,~)=e(N~, ~)-r=e(N~)+ l-r=i(T~)+ l. []
2. T h e global case
Let Yl=Quot(E, l ) x X , and consider the universal exact sequence of sheaves
Oil 17//,
0
) . A / - - ~ OQuot @ E ~
T
70.
The function y = (s, x)Hi(T~,x) is upper semi-continuous. Let Yl# denote the locally
closed subset { y = (s, x)EYI li(T~,x)=i} with the reduced subscheme structure.
P r o p o s i t i o n 5. The scheme Yl is irreducible of dimension (r+l)l+2. For
each i>O one has codim(Yl,i,Yz)_>2i.
Clearly, the first assertion of the theorem follows fi'om this.
Pro@ The proposition will be proved by induction on l, the case l=1 being
trivial: Y x = P ( E ) x X , the stratum Y~,I is the graph of the projection P ( E ) ~ X
and Yl,i 0 for i>_2. Hence suppose the proposition has been proved for some l_>l.
We describe the 'global' version of the elementary modification discussed above.
Let Z = P ( I V ) be the projectivization of the family iV and let ~ = ( ~ 1 , P~): Z-~
Yl=Quot(E,l) x X denote the natural projection morphism. On Z x X there is a
canonical epimorphism
A: (Pl x idx)*iv ~
(idz, P2).~*iV----* (idz, F2).Oz (1) =:/~.
As before we define a family T' of quotients of length l + l by means of A,
0
0--
--->]C
> (Fl,idx)*iV
> T'
> (~l,idx)*T
~0
> Oz@E
~ (Fl,idx)*T
> 0.
Irreducibility of t h e p u n c t u a l quotient scheme of a surface
249
Let r Z ~ Q u o t ( E , / + 1 ) be the classifying morphism for the family T', and define
"~:=(~1, ~2:=~2): Z ~ Y I + I . The scheme Z together with the morphisms p: Z-~Yz
and ~: Z--~Yt+I allows us to relate the strata ~,i and Yz+lj. Note that ~ ( Z ) =
Uj>a Y/+I,j 9
The fibre of qD over a point (s,x)EYl# is given by P(2V's(x))~P r-a+i, since
dim(AY~(x))=r+i(T~,~)=r+i by Lemma 2. Similarly, the fibre of ~ over a point
(s',x)CYZ+l,j is given by P(Soc(T~, x ) V ) ~ P j - t . If T' is obtained from T by an
elementary modification, then li(T')-i(T)I_< 1 a s shown above. This can be stated
in terms of ~ and ~ as follows: For each j >_1 one has
U
Ii-jl_<l
Using the induction hypothesis on the dimension of Yz,i and the computation of the
fibre dimension of ~ and ~, we get
d i m ( Y ~ + l , j ) + ( j - 1 ) _< max
Ii jl_<l
{(r+l)l+2-2i+(r-l+i)}
and
dim(Yz+lj) <
'
-
(r+l)(l+l)+2-2j- li-jI_<l
rain { i - j + l } .
As minli_jk<1{i-j+l}_>O , this proves the dimension estimates of the proposition.
It sufficesto show that Z is irreducible. Then Quot(E, l+l)=~1(Z) and ~ + I
are irreducible as well.
Since X is a smooth surface, the epimorphism OQuot QE--+q- can be completed
to a finite resolution
0 -----~A ----+ B
>OQuotGE
>T - - - + 0
with locally free sheaves .4 and B on Yz of rank n and n+r, respectively, for some
positive integer n. It follows that Z = P ( N ) c P ( B )
is the vanishing locus of the
composite homomorphism ~* A-~ ~* B ~ Op(s) (1). In particular, assuming by induction that Yz is irreducible, Z is locally cut out from an irreducible variety of
dimension ( r + l ) / + 2 + ( r + n - 1 )
by n equations. Hence every irreducible component of Z has dimension at least ( r + l ) ( l + l ) . But the dimension estimates for the
stratum Y~,i and the fibres of ~ over it yield
dim(~-l(Yl,i)) < ( r + 1 ) / + 2
2i+(r+i--1) = (r+l)(l+l)-i,
which is strictly less than the dimension of any possible component of Z, if i > 1. This
implies that the irreducible variety ~-1 (Y1,0) is dense in Z. Moreover, since the fibre
o f r over Y~+1,1 is zero-dimensional, dim(Yl+~)=dim(Y~+l,1)+2=dim(Z)+2 has the
predicted value. []
250
Geir Ellingsrud and Manfred Lehn
3. T h e local case
We now concentrate on quotients E--~T, where T has support in a single fixed
closed point x E X . For those quotients the structure of E is of no importance, and
we may assume that E~--O~x . Let Q[ denote the closed subset
{[O k ~ T] C Quot(O~, l) I Supp(T ) : {x}}
with the reduced subscheme structure. We may consider Q[ as a subscheme of Y41
by sending [q] to ([q],x). Then it is easy to see that ~-1 ( Qrl ) = ~ -1 (Q~z+l)- Let this
scheme be denoted by Z ~.
We will use a stratification of Q[ both by the socle dimension i and the number
of generators e of T and denote the corresponding locally closed subset by Q~'~
1,i 9
Moreover, let Q}',i=U~ Qr'r
1,i and define Q~,e
l similarly. Of course, Qr'r
1,i is empty
unless l < i < l and l < e < m i n { r , 1}.
To prove the second half of the theorem it suffices to show the following.
P r o p o s i t i o n 6. The scheme Q[ is an irreducible variety of dimension r l - 1.
L e m m a 7. We have dim(Q1, i )<_ ( r l - 1 ) - ( 2 ( i - 1)+ (~)).
Proof. The proof is done by induction on l. If l = 1, then '~lDr =.L~'~Dr--1 , and QI',~=0
if e>2 or i_>2. Assume that the lemma has been proved for some I>1.
Let [q':O~x-*T '] cQl+l,
~'~ j be a closed point. Suppose that the map #: k(z)--~
T ' ( z ) represents a point in ~b-1 ([q']) =P(Soc(T~) v) and that T~ =coker(#) is the corresponding modification. If i=i(T~,x) and c=e(T~,x), then, according to Section 1,
the pair (i, r can take the following values
(i)
(i,c)=(j-l,e-1),
(j-l,e),
(j,e) or ( j + l , e ) ,
in other words
--1
r,e
--1 r,e 1
(Ql+l,J)C~) (Ql,J -1)U
- l / Q r e,
( l'i)"
U (p
li-j[<l
Subdivide A -Dr'~-~4j into four locally closed subsets Ai,~ according to the generic
value of (i, c) on the fibres of ~. Then
dim(A~,~) + ( j - 1) < dim(Q1
. . . . i ) +di,~,
where di,~ is the fibre dimension of the morphism
~:r
-1 A
N -1 r,~
r,e
(~,~) ~ ( Q 4 i ) - - ~ Q 4 i "
Irreducibility of the punctual quotient scheme of a surface
9
251
r~E
By the induction hypothesis we have bounds for dlm(Q~, i ), and we can bound di,~
in the four cases (1) as follows9
~r,e-- i
(A) Let [q:O~x--*T]eql,j_ 1 be a closed point with N = k e r ( q ) . As we are looking
for modifications T ' with e(Ttx)=e, we are in the situation of L e m m a 3 and may
conclude
~ - 1 ([q])Nr
since
(Ae-l,j-1)
im(k(x)T--~T(x))~-k ~
1. Hence
9
dim(Aj_<~_ J _< dim
TM
N(x) ~ k(x)r))
P(ker(k(x) ~ --* T(x) ) ) ~ P~-~,
P(im(g:
dj_l,e_l=r-e
Qt,'j-1 +(r-e) - (jre--1
and
1)
<_{(rl-l)-2(j-2)-(e21)}+(r-e)-(j-1)
={(r(l+l)-l)-2(j-l)-(:)}-(j-2) 9
Note t h a t this case only occurs for j > 2 , so that ( j - 2 ) is always nonnegative.
(B) In the three remaining cases
e=e
and
i=j-1, j,
orj+l,
we begin with the rough estimate di,~ <_r+i-1 as in Section 2. This yields
dim(A<~) <
{(rl-1)-2(i-1)-(~)}+(r+i-1)-(j-1)
(2)
={(r(l+X)-l)-2(j-1)-(:)}-(i-j).
Thus, if i=j we get exactly the estimate asserted in the lemma, if i = j + l the
estimate is better t h a n what we need by 1, but if i = j - 1 , the estimate is not
good enough and fails by 19 It is this latter case t h a t we must further study9 Let
[q: O~c --~T] be a point in QT,~
<j-1 with N = k e r ( q ) . There are two alternatives.
(i) Either the fibre )9-1 ([q])N• - I ( A j 1,~) is a proper closed subset of P ( N ( x ) )
which improves the estimate for the dimension of the fibre qo-1 ([q]) by 1;
(ii) or this fibre equals with P ( N ( x ) ) , which means t h a t the socle dimension
increases for all modifications of T. In this case we conclude from L e m m a 4 that
also i(T-) =i(T) + 1 for every modification T - = coker (#-: k (x) --~T). But, as we just
saw, calculation (2), applied to the contribution of Qz-l,j to Qz,j-1, shows that the
dimension estimate for the locus of such points [q] in Q~'~
1,j-1 can be improved by 1
compared to the dimension estimate for Q~'~
t,j-1 as stated in the lemma.
252
Geir Ellingsrud
and Manfred
Lehn
Hence in either case we can improve estimate (3) by 1 and get
dim(Aj 1,~)_< (r(l+ 1 ) - 1 ) - 2 ( j - 1 ) - ( ~ )
as required. Thus, the lemma holds for l + l .
L e m m a 8.
We have ~(~
[]
(Ql ))cQt+l"
P r o 4 Let [q: (9)-~r] cQ[': be a closed point with N=ker(q). Then s 1([@ =
P ( N ( x ) ) ~ P ~+i-1 and ~ l([q])C~r
Since we always have e > l , i>1, a dense open part of p-~([q]) is mapped to Q~'~
[]
--
Lemma
--
l+l"
9. If r>_2 and if Q[-~ is irreducible of dimension ( r - i ) / - 1 ,
Q~,<,
l
: = ~j,< < Q .l
.
. subset of Ql of dimension r l - 1 .
is an. irreducible
open
then
Pro@ Let M be the variety of all r x ( r - 1) matrices over k of maximal rank,
and let 0 ~ O K- 1~ O~--* s ~ 0 be the corresponding tautological sequence of locally
free sheaves on M. Consider the open subset U C M x Q[ of points (A, [O ~~ T ] ) such
that the composite homomorphism
0 r-I ~
O r
)T
is surjective. Clearly, the image of U under the projection to Q[ is Qr,<r
l
' On the
other hand, the tautological epimorphism
r--i
Ov• x
r
) Ov• x --~ (OM|215
induces a classifying morphism g': U - ~ r" ~vl ' - I
9:(prl,gl):U
"
x
The morphism
, Mx
Q?'--I
l
is surjeetive. In fact, it is an a n n e fibre bundle with fibre
g ~(g(A, [O ~-1 ~ r ] ) ) ~I~om~(C(A) , T ]~---~A
~
k"
Since Qr-I~ is irreducible of dimension ( r - 1 ) l - 1 by assumption, U is irreducible of
dimension r l - l + d i m ( M ) , and Q<'<~
is irreducible of dimension rl 1. []
1
Proof of Proposition 6. The irreducibility of Q[ will be proved by induction
over r and 1. The case /=1, r arbitrary is trivial; whereas the case l arbitrary,
r = l is the case of the Hilbert scheme, for which there exist several proofs ([2], [5]).
I r r e d u c i b i l i t y of t h e p u n c t u a l q u o t i e n t s c he me of a surface
253
Assume therefore that r_>2 and that the proposition holds for (1, r) and (l+ 1, r - 1 ) .
We will show that it holds for (l + 1, r) as well.
Recall that Z r : p -1 ( Q rz ) - Q l r x ~ Z. Every irreducible component of Z' has dimension greater than or equal to d i m ( Q ~ ) + r 1 = r ( / + 1 ) - 2 (cf. Section 2). On the
other hand, dim(p-l(Q~,i))<rI-1-2(i-1)+(r+i 1)=r(l+l)-i. Thus an irreducible component of Z' is either the closure of ~r~
~-1/Qr
\ 1,1)~ (of dimension r(l+ 1 ) - 1 )
or the closure of p - 1 (W) for an irreducible component W C Qt~,2 of maximal possible
dimension rl-3. But according to Lemma 8 the image of p - I ( w ) under ~b will be
r, <r
contained in the closure of D
"~Z+l
, unless W is contained in Q~'~
1,2- But Lemma 7
says that Qr,~ has codimension > 2 + ( ; ) > 3 if r > 2 , and hence cannot contain W for
dimensional reasons. Hence any irreducible component of Z ~ is mapped by ~p into
the closure of r)~'<~
" ~ l + l which is irreducible by Lemma 9 and the induction hypothesis.
This finishes the proof of the proposition. []
254
Geir Ellingsrud and Manfred Lehn:
Irreducibility of the punctual quotient scheme of a surface
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AG/9703038.
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3. IARROBINO, A., Reducibility of the families of 0-dimensional schemes on a variety,
Invent. Math. 15 (1972), 72-77.
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5. ELLINGSRUD~ C-. and 8TROMME, S. A., An intersection number for the punctual
Hilbert scheme, Trans. Amer. Math. Soc. 350 (1998), 2547-2552.
6. FOGARTY, J., Algebraic families on an algebraic surface, Amer. J. Math. 90 (1968),
511 521.
7. GIESEKER, D. and LI, J., Moduli of high rank vector bundles over surfaces, J. Amer.
Math. Soc. 9 (1996), 107-151.
8. GROTHENDIECK~ A., Techniques de construction et th~or~mes d'existence en g~om~trie algSbrique IV: Les sch@mas de Hilbert, in Sdminaire Bourbaki Volume 1960//1961, Ezposd 221, pp. 249 276, Benjamin, New York-Amsterdam,
1966.
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Differential Geom. 37 (1993), 417-466.
Received April 9, 199Y
in revised form August 18, 1998
Geir Ellingsrud
Department of Mathematics
University of Oslo
P.O. Box 1053
NO-0316 Oslo
Norway
emaih [email protected]
Manfred Lehn
Mathematisches Institut
Georg-August-Universit/it
Bunsenstrasse 3-5
DE-37073 GSttingen
Germany
emaih [email protected]